Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given is an element of > 0, log L(r, f) > (1- is an element of) log M(r, f) for all r outside a set of logarithmic density zero. Let F = U-K >= (1) F-K where FK is the set of all transcendental entire functions f such that log log M(r, f) >= (log r) 1/K. If h = f(N) o f(N-1) o... o f(1) where f(i) is an element of F boolean AND L (i = 1,..., N), then it is shown that h has no unbounded Fatou component. (c) 2007 Elsevier Inc. All rights reserved.